Prove that if than . ( )

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- Nov 15th 2007, 07:11 AMjames_bondInequality
Prove that if than . ( )

- Nov 16th 2007, 11:28 AMOpalg
Who thinks these horrible things up (and why do I spend hours brooding over them)?

I'll assume that a, b and c are all positive (the problem's hard enough as it is, and I don't want to be bothered with negative quantities).

Notice first that (and similarly for b and c), so the inequality can be written

. . . . . . . .**(1)**If bc+ca+ab=1 then

. . . . . . . .**(2)**

(with similar inequalities for b and c).

Also, . . .**(3)**

(just multiply out both sides to verify that).

By the arithmetic-geometric mean inequality,

. . . . . . . .**(4)**

(that's the only place where I assume that a, b and c are positive).

It follows that

. . . . . . . .by**(2)**. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .by**(3)**. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .by**(4)**.

By**(1)**, that was what we wanted to prove.